A306612 - OEIS

%I #46 Sep 04 2019 12:11:43

%S 3,4,7,8,16,16,157,157,16957,19231,80942,82372,82372,9624266,19607227,

%T 118867612,4968215191,31090893772,118903377091,187341482252,

%U 1784664085208,12330789708022,68016245854132,68016245854132,10065964847743822,74887595879692807,1825207861455319267,98403562254816509476,283462437415903129597,2126598918934702375802

%N a(n) is the least integer k > 2 such that the remainder of -k modulo p is strictly increasing over the first n primes.

%C 0, 1, and 2 satisfy this condition for all p, so this sequence starts at 3. The growth of this sequence is much more irregular than that of the companion sequence A306582.

%e    a(n) modulo 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

%e   ===== ==================================================

%e       3        1, 0, 2, 4,  8, 10, 14, 16, 20, 26, 28, ...

%e       4        0, 2, 1, 3,  7,  9, 13, 15, 19, 25, 27, ...

%e       7        1, 2, 3, 0,  4,  6, 10, 12, 16, 22, 24, ...

%e       8        0, 1, 2, 6,  3,  5,  9, 11, 15, 21, 23, ...

%e      16        0, 2, 4, 5,  6, 10,  1,  3,  7, 13, 15, ...

%e     157        1, 2, 3, 4,  8, 12, 13, 14,  4, 17, 29, ...

%e   16957        1, 2, 3, 4,  5,  8,  9, 10, 17,  8,  0, ...

%e   19231        1, 2, 4, 5,  8,  9, 13, 16, 20, 25, 20, ...

%e   80942        0, 1, 3, 6,  7,  9, 12, 17, 18, 26, 30, ...

%o (PARI) isok(k, n) = {my(last = -1, cur); for (i=1, n, cur = -k % prime(i); if (cur <= last, return (0)); last = cur;); return (1);}

%o a(n) = {my(k=3); while(!isok(k, n), k++); k;} \\ _Michel Marcus_, Jun 04 2019

%o (Python)

%o from sympy import prime

%o def A306612(n):

%o     plist, x = [prime(i) for i in range(1,n+1)], 3

%o     rlist = [-x % p for p in plist]

%o     while True:

%o         for i in range(n-1):

%o             if rlist[i] >= rlist[i+1]:

%o                 break

%o         else:

%o             return x

%o         for i in range(n):

%o             rlist[i] = (rlist[i] - 1) % plist[i]

%o         x += 1 # _Chai Wah Wu_, Jun 15 2019

%Y Cf. A306582.

%K nonn,hard

%O 1,1

%A _Charlie Neder_, Jun 03 2019

%E a(16)-a(19) from _Daniel Suteu_, Jun 04 2019

%E a(20)-a(25) from _Giovanni Resta_, Jun 16 2019

%E a(26)-a(27) from _Bert Dobbelaere_, Jun 22 2019

%E a(28)-a(30) from _Bert Dobbelaere_, Sep 04 2019